A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 6, 1, 0, 1, 11, 18, 1, 0, 1, 30, 75, 40, 1, 0, 1, 52, 420, 350, 75, 1, 0, 1, 126, 1218, 3080, 1225, 126, 1, 0, 1, 219, 4242, 17129, 15750, 3486, 196, 1, 0, 1, 510, 14563, 82488, 152355, 63756, 8526, 288, 1, 0, 1, 896, 42930, 464650, 1049895, 954387, 217560, 18600, 405, 1

Offset: 0

Author

Alois P. Heinz, Apr 27 2017

Keywords

• nonn
• tabl

Examples

			T(3,1) = 1: 123.
T(3,2) = 6: 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
T(3,3) = 1: 1|2|3.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   6,    1;
  0, 1,  11,   18,     1;
  0, 1,  30,   75,    40,     1;
  0, 1,  52,  420,   350,    75,    1;
  0, 1, 126, 1218,  3080,  1225,  126,   1;
  0, 1, 219, 4242, 17129, 15750, 3486, 196, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-10 give: A000007, A057427, A285917, A285918, A285919, A285920, A285921, A285922, A285923, A285924, A285925.

Main diagonal and first lower diagonal give: A000012, A002411.

Row sums give A120774.

T(2n,n) gives A285926.

Cf. A048993, A131689, A226874, A285849.

Programs

Maple

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*x^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n-i*j, i-1, p+j]*x^j*multinomial[n, Join[{n-i*j}, Table[i, j]]]/ j!^2, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)